From the Preface (1955): The first part of the present exposition is devoted to the theory of Toeplitz forms. The second part deals with applications, in particular to the calculus of probability and mathematical statistics. Neither part claims completeness in any way. Our purpose has been to elucidate the principal ideas of this remarkable chapter of modern analysis and to help the interested student of mathematical statistics to acquire a working knowledge of the subject. The somewhat protracted Chapter 1 explains not only the notation employed but contains also the definition of important auxiliary concepts and the exposition of basic results which will be used later. This arrangement avoids interruptions in the main text ... [It is assumed] that the reader is in possession of the fundamental facts of the theory of functions.

"In chapters 2 and 3 certain topics appear which were treated in the book on orthogonal polynomials by G. Szegö. In view of the progress made in this subject since the publication of that book (1939) it was possible to bring some details in an improved setting. The other chapters contain partly old and partly more recent results, some older facts in a new setting, and finally some completely new results. Chapter 1-6 and Chapter 9 have been prepared by Szegö, the other chapters by Grenander ... "

Readership

Graduate students and research mathematicians.

Table of Contents

• Preliminaries
• Orthogonal polynomials. Algebraic properties
• Orthogonal polynomials. Limit properties
• The trigonometric moment problem
• Eigenvalues of Toeplitz forms
• Generalizations and analogs of Toeplitz forms
• Further generalizations
• Certain matrices and integral equations of the Toeplitz type

• Applications to analytic functions
• Applications to probability theory
• Applications to statistics
• Appendix: Notes and references
• Bibliography
• Index